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Perfect Packings in Quasirandom Hypergraphs

John LenzJoint work with Dhruv Mubayi

University of Illinois at Chicago

June 13, 2013

John Lenz Perfect Packings in Quasirandom Hypergraphs

Quasirandom Graphs

Fix 0 < p < 1. Let G = Gnn→∞ be a sequence of graphs with|V (Gn)| = n and |E (Gn)| = p

)+ o(n2).

G satisfies Discp if ∀U ⊆ V (Gn),

e(Gn[U]) = p

)+ o(n2)

G satisfies Countp[All] if for all graphs F , the number oflabeled copies of F in Gn is

p|E(F )|n|V (F )| + o(n|V (F )|).

Theorem (Thomason 1987, Chung-Graham-Wilson 1989)

Discp and Countp[All] are equivalent.

Quasirandom Graphs

)+ o(n2).

e(Gn[U]) = p

)+ o(n2)

p|E(F )|n|V (F )| + o(n|V (F )|).

Quasirandom Graphs

)+ o(n2).

e(Gn[U]) = p

)+ o(n2)

p|E(F )|n|V (F )| + o(n|V (F )|).

Quasirandom Graphs

)+ o(n2).

e(Gn[U]) = p

)+ o(n2)

p|E(F )|n|V (F )| + o(n|V (F )|).

Quasirandom Hypergraphs

Observation (Rodl)

For 3-uniform hypergraphs, Disc1/4 6⇔ Count1/4[All]

Proof.

Use Erdos and Hajnal’s construction: let T be a random graphtournament and form a three-uniform hypergraph by making each

cyclically oriented triangle a hyperedge. There is no K(3)4 but

Disc1/4 holds.

Quasirandom Hypergraphs

Observation (Rodl)

For 3-uniform hypergraphs, Disc1/4 6⇔ Count1/4[All]

Proof.

Use Erdos and Hajnal’s construction: let T be a random graphtournament and form a three-uniform hypergraph by making each

cyclically oriented triangle a hyperedge. There is no K(3)4 but

Disc1/4 holds.

Perfect Packings in Graphs

Definition

Let G and F be graphs or k-uniform hypergraphs. We say that Ghas a perfect F -packing if the vertices of G can be covered byvertex disjoint copies of F .

Theorem (Hajnal-Szemeredi 1970)

If r divides n = |V (G )| and δ(G ) ≥ (1− 1/r)n, then G contains aperfect Kr -packing.

Theorem (Komlos-Sarkozy-Szemeredi 2001 – Alon-Yuster Conjecture)

For every F and G where |V (F )| divides n = |V (G )| andδ(G ) ≥ (1− 1/χ(F ))n + CF , G contains a perfect F -packing.

Definition

Perfect Packings in Hypergraphs

Theorem (Rodl-Rucinski-Szemeredi 2009, Kuhn-Osthus 2006)

If H is a k-uniform hypergraph, k divides n = |V (H)|, andδcodeg (H) ≥ n/2− k + C , then H has a perfect matching whereC ∈ 3/2, 2, 5/2, 3 depends on the values of n and k.

Other results for various hypergraphs F are known, including K4

(Keevash-Mycroft, Lo-Markstrom, Pikhurko), K−4 (Lo-Markstrom),K4 − 2e (Kuhn-Osthus, Czygrinow-DeBiasio-Nagle)

Perfect Packings in Hypergraphs

Theorem (Rodl-Rucinski-Szemeredi 2009, Kuhn-Osthus 2006)

If H is a k-uniform hypergraph, k divides n = |V (H)|, andδcodeg (H) ≥ n/2− k + C , then H has a perfect matching whereC ∈ 3/2, 2, 5/2, 3 depends on the values of n and k.

Other results for various hypergraphs F are known, including K4

(Keevash-Mycroft, Lo-Markstrom, Pikhurko), K−4 (Lo-Markstrom),K4 − 2e (Kuhn-Osthus, Czygrinow-DeBiasio-Nagle)

Quasirandomness and Perfect Packings

Let G = Gnn→∞ be a sequence of graphs. We say that G has aperfect F -packing if all but finitely many of the graphs Gn with|V (F )| dividing n have a perfect F -packing.

Theorem (Komlos-Sarkozy-Szemeredi 1997)

Let 0 < p < 1 be fixed and let F be any graph. Let G be a graphsequence satisfying Discp with δ(Gn) = Ω(n). Then G has aperfect F -packing.

Problem

Characterize the 3-uniform hypergraphs F for which for all0 < p < 1 a hypergraph sequence H satisfying Discp withδ(Hn) = Ω(n2) is forced to have a perfect F -packing.

Quasirandomness and Perfect Packings

Let G = Gnn→∞ be a sequence of graphs. We say that G has aperfect F -packing if all but finitely many of the graphs Gn with|V (F )| dividing n have a perfect F -packing.

Theorem (Komlos-Sarkozy-Szemeredi 1997)

Let 0 < p < 1 be fixed and let F be any graph. Let G be a graphsequence satisfying Discp with δ(Gn) = Ω(n). Then G has aperfect F -packing.

Problem

Characterize the 3-uniform hypergraphs F for which for all0 < p < 1 a hypergraph sequence H satisfying Discp withδ(Hn) = Ω(n2) is forced to have a perfect F -packing.

Some Hypergraphs

A hypergraph F is linear if every pair of distinct edges shareat most one vertex.

Cherry (2, 1)-four-cycle

Our Results - Perfect Packings

Theorem (L-Mubayi)

Fix 0 < p < 1. Let H be a 3-uniform hypergraph sequencesatisfying Discp. Then H has a perfect F -packing if

F is linear and δ(Hn) = Ω(n2),

F is the cherry and δcodeg (Hn) = Ω(n),

F is the (2, 1)-four-cycle and δcodeg (Hn) = Ω(n).

Theorem (L-Mubayi)

Our Results - Constructions

The Erdos-Hajnal construction satisfies Disc1/4, has minimum

codegree (1− o(1))n4 , and has no perfect K−4 -packing.

Theorem (L-Mubayi)

There exists a 3-uniform hypergraph sequence H satisfying Disc1/8with δcodeg (Hn) ≥ (1− o(1))n8 and has no perfect cherry-four-cyclepacking. The cherry four cycle is the following hypergraph:

Our Results - Constructions

The Erdos-Hajnal construction satisfies Disc1/4, has minimum

codegree (1− o(1))n4 , and has no perfect K−4 -packing.

Theorem (L-Mubayi)

There exists a 3-uniform hypergraph sequence H satisfying Disc1/8with δcodeg (Hn) ≥ (1− o(1))n8 and has no perfect cherry-four-cyclepacking. The cherry four cycle is the following hypergraph:

Sparse Setting

Theorem (Krivelevich-Sudakov 2002)

If G is a regular, n-vertex graph with

λ2(G ) ≤ (log log n)2

1000 log n (log log log n)λ1(G )

and n is large, then G is Hamiltonian.

Eigenvalue definitions of Friedman and Wigderson

Let H be a 3-uniform, n-vertex hypergraph. The adjacencymap of H is

τ : Rn × Rn × Rn → R

τ(ex , ey , ez) =

1 if xyz ∈ E (H)

0 otherwise

λ1(H) := supτ(w ,w ,w) : w ∈ Rn, ‖w‖ = 1.Let J : Rn × Rn × Rn → R be the all-ones map.

λ2(H) := supx∈Rn

‖x‖=1

∣∣∣∣τ(x , x , x)− k!|E (H)|nk

J(x , x , x)

∣∣∣∣

τ(ex , ey , ez) =

1 if xyz ∈ E (H)

0 otherwise

λ1(H) := supτ(w ,w ,w) : w ∈ Rn, ‖w‖ = 1.

Let J : Rn × Rn × Rn → R be the all-ones map.

λ2(H) := supx∈Rn

‖x‖=1

∣∣∣∣τ(x , x , x)− k!|E (H)|nk

J(x , x , x)

∣∣∣∣

τ(ex , ey , ez) =

1 if xyz ∈ E (H)

0 otherwise

λ1(H) := supτ(w ,w ,w) : w ∈ Rn, ‖w‖ = 1.Let J : Rn × Rn × Rn → R be the all-ones map.

λ2(H) := supx∈Rn

‖x‖=1

∣∣∣∣τ(x , x , x)− k!|E (H)|nk

J(x , x , x)

∣∣∣∣

Theorem (L-Mubayi)

Let H be a 3-uniform, n-vertex hypergraph with n large and letp = |E (H)|/

). If n is divisible by three, δcodeg (H) ≥ pn

100 , and

λ2(H) ≤ Cp15/2λ1(H),

then H contains a perfect matching.

Absorbing Sets

To prove the existence of perfect F -packings, we use theabsorption method of Rodl-Rucinski-Szemeredi.

Definition

Let F and G be graphs. A vertex set A ⊆ V (G ) F -absorbs a setB ⊆ V (G ) if both G [A] and G [A ∪ B] have perfect F -packings.

A K3-absorbs B

Absorbing Sets

To prove the existence of perfect F -packings, we use theabsorption method of Rodl-Rucinski-Szemeredi.

Definition

Let F and G be graphs. A vertex set A ⊆ V (G ) F -absorbs a setB ⊆ V (G ) if both G [A] and G [A ∪ B] have perfect F -packings.

A K3-absorbs B

Open Problems

Characterize the hypergraphs F for which Discp and linearmin codegree imply a perfect F -packing.

Does Discp and linear min codegree imply the existence ofspanning structures? Hamilton cycles? Any linear hypergraph?

For K4 and the cherry four-cycle, what values of p causeDiscp and linear min codegree to imply a perfect packing?

What about other hypergraph quasirandom properties besidesDiscp? What can they pack?

Open Problems

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